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53

The Hamiltonian H and Lagrangian L which are rather abstract constructions in classical mechanics get a very simple interpretation in relativistic quantum mechanics. Both are proportional to the number of phase changes per unit of time. The Hamiltonian runs over the time axis (the vertical axis in the drawing) while the Lagrangian runs over the trajectory of ...


41

Here's what I perceive to be a mathematically and logically precise presentation of the theorem, let me know if this helps. Mathematical Preliminaries First let me introduce some precise notation so that we don't encounter any issues with "infinitesimals" etc. Given a field $\phi$, let $\hat\phi(\alpha, x)$ denote a smooth one-parameter family of fields ...


29

In physics, it is often implicitly assumed that the Lagrangian $L=L(\vec{q},\vec{v},t)$ depends smoothly on the (generalized) positions $q^i$, velocities $v^i$, and time $t$, i.e. that the Lagrangian $L$ is a differentiable function. Let us now assume that the Lagrangian is of the form $$L~=~\ell\left(v^2\right),\qquad\qquad v~:=~|\vec{v}|,\tag{1}$$ where $...


26

Setting. We are considering a transformation, which may transform the field variables $\phi^{\alpha}(x)$ and which may transform the space-time points $x^{\mu}$. The transformation in turn apply to The action $S_V[\phi]=\int_V \! d^nx~{\cal L} $. The Euler-Lagrange equations = the equations of motion (EOM). A solution $\phi$ of EOM. Definition. If any of ...


20

As that lovely article linked by dfan says the virial theorem comes from varying the action $S[x]$ by $x\rightarrow(1+\epsilon)x$ $$\frac{1}{T}\delta S = \frac{1}{T}\epsilon\int_{0}^{T} dt\{m\dot{x}^2 -x\frac{\partial V}{\partial x}\}$$ This is a variation of the action and therefore must vanish up to some boundary terms if $x$ is a solution of the ...


20

I) Initial value problems and boundary value problems are two different classes of questions that we can ask about Nature. Example: To be concrete: an initial value problem could be to ask about the classical trajectory of a particle if the initial position $q_i$ and the initial velocity $v_i$ are given, while a boundary value problem could be to ask ...


18

The action functional and Hamilton's principal function are two different mathematical objects related to the same physical quantity. The action along a trajectory $\gamma:[t_1,t_2]\rightarrow Q$ is given by $$ S[\gamma] = \int_{t_1}^{t_2}L(\gamma(t'),\dot\gamma(t'),t')dt' $$ whereas the principal function is the solution of the Hamilton-Jacobi equation $$ ...


16

Some time after Newton described the laws of nature in terms of an instantaneous relationship, others noticed that the history, rather than the instantaneous state, of a system could, in at least one case, be described by saying that it obeyed a certain relationship: a particular function describing the history must always be the one that (a) starts and ends ...


16

What would come the most close to that would be Julian Schwinger's Quantum Action Principle, see for example this pdf: https://arxiv.org/abs/1503.08091. Or see his paper "the Theory of quantized Fields I". What he essentially does is, he promotes the Action to be an Operator, and states that the variation of this Operators matrix elements equals the ...


15

I) Not all equations of motion (eom) are variational. A famous example is the self-dual five-form in type IIB superstring theory. In classical point mechanics, frictional forces typically lead to non-variational problems. II) Consider for instance $n$ variable $q^i$ and $n$ eoms, $$\tag{1} E_i~\approx~ 0, \qquad i~\in~\{1, \ldots, n\}. $$ A simplified ...


14

I) At least three different quantities in physics are customary called an action and denoted with the letter $S$. The (off-shell) action $$\tag{1}S[q]~:=~ \int_{t_i}^{t_f}\! dt \ L(q(t),\dot{q}(t),t) $$ is a functional of the full position curve/path $q^i:[t_i,t_f] \to \mathbb{R}$ for all times $t$ in the interval $[t_i,t_f]$. See also this question. (...


14

Quantum systems are essentially defined by their symmetries. For example, in QFT's you expect all terms not forbidden by the symmetries of the problem to appear in the Lagrangian, with irrelevant operators suppressed by large scales, etc. So I think your first step in this approach would be to write down the most general 2D QFT respecting the 2D Diff and ...


14

Perhaps a simple example is in order. Consider a harmonic oscillator $$\tag{1} S~=~\int_{t_i}^{t_f} \! dt~L, \qquad L~=~\frac{m}{2}\dot{x}^2 - \frac{k}{2}x^2, $$ with characteristic frequency $$\tag{2} \frac{2\pi}{T}~=~\omega~=~\sqrt{\frac{k}{m}}, $$ and Dirichlet boundary conditions $$\tag{3} x(t_i)~=~x_i \quad\text{and}\quad x(t_f)~=~x_f. $$ For ...


14

We do not treat $\dot q$ as an independent variable in the derivation of the Euler-Lagrange equations. The rough answer is that $q$ and $\dot q$ are independent as inputs to the Lagrangian, but become linked once we specify a path through configuration space - I expand on this in points 5 and 6. I'll be quite formal in what follows, but perhaps the ...


13

The dimensions of the Planck constant $\hbar$, the action $S$, and the angular momentum, are constrained by the following important facts: A conjugate pair of two observables is quantum mechanically related to the Planck constant $\hbar$ via a Heisenberg uncertainty relation. A conjugate pair of two variables is classically related to the action $S$ ...


13

First I want to remind you what is going on behind the scenes. You know where the particle is at some initial time $t_1$, and you know where the particle is at some final time $t_2$, and the question you are asking is, which path will get me from the initial position at the initial time to the final position at the final time in a way that minimizes the ...


13

I) The closest cosmetic resemblance between the Nambu-Goto action and the Polyakov action is achieved if we write them as $$\tag{1} S_{NG}~=~ -\frac{T_0}{c} \int d^2{\rm vol} ~\det(M)^{\frac{1}{2}} , $$ and $$\tag{2} S_{P}~=~ -\frac{T_0}{c}\int d^2{\rm vol}~ \frac{{\rm tr}(M)}{2} , $$ respectively. Here $h_{ab}$ is an auxiliary world-sheet (WS) metric ...


13

We want to calculate the path integral $$ Z = \int \mathcal{D}{\phi}\, e^{i \hbar^{-1} S[\phi]} $$ which encodes a transition amplitude between initial and final quantum states. If we had the effective action $\Gamma[\phi]$ at our disposal, we would have calculated the same result by solving for $$ \phi_c(x):\quad \left. \frac{\delta \Gamma}{\delta \phi} \...


13

The problem with your approach is that your proposed action $$S = \int |\mathbf{v}| \, dt$$ is not invariant at all. While Landau's action is invariant under Lorentz transformations, and in fact completely coordinate independent, yours is not invariant under even Galilean transformations, which add a constant to $\mathbf{v}$. The space-only analogue of a ...


12

Dear Ondřeji, a good question but a part of the answer is that your equation for the fluid is underdetermined. It treats $p,\rho$ as independent variables. But the physical system only knows how to behave if you also substitute some equation of state, i.e. a function $p=p(\rho)$ or $p=p(\rho,\vec v)$. Note that your Ansatz for the stress-energy tensor ...


12

Yes, the invariance of the action follows from special relativity – and special relativity is right (not only) because it is experimentally verified. All the equations of motion may be derived from the condition $\delta S = 0$, the action is stationary (which usually means it has the minimum value on the allowed trajectory/history among all trajectories/...


12

Here we will assume that we ultimately want to consider the full quantum theory, usually written in terms of a gauge-fixed path integral $$Z~=~\int \!{\cal D}\phi~ \exp\left(\frac{i}{\hbar}S_{\rm gf}[\phi]\right) \tag{1}$$ rather than just the classical action and the corresponding classical equations of motion (with or without gauge-fixing terms). If the ...


12

Comment to the question (v2): P&S is using the notation of a 'same-spacetime' functional derivative. To illustrate this notation, let us for simplicity stay within first variations, and leave it to the reader to generalize to higher-order variations. I) First of all, functional/variational derivatives should not be confused with partial derivatives. In ...


12

Dirac is indeed the forefather of the path integral approach to quantum mechanics. His reasoning was described in the 1933 paper "Lagrangian in/of quantum mechanics". See the full text: http://www.ifi.unicamp.br/~cabrera/teaching/aula%2015%202010s1.pdf Conceptually, he had the whole thing. He realized that the Poisson brackets have a counterpart in ...


12

Setting the einbein to $1$ corresponds to a diffeomorphism of the metric, as the einbein is given by $e_{\tau\tau}=\sqrt{g_{\tau\tau}}$, which can be easily deduced from the fact that a the vielbein is given as the transformation coefficients from the coordinate basis to a non-coordinate basis. Hence, the dimensionality of the einbein depends on that of the ...


12

The Lagrangian and Hamiltonian approaches are frameworks, and not theories. It is certainly true that a wide variety of systems are susceptible to such an approach. However, there are many theories which do not possess Lagrangians. For example, it is believed that a certain set of six-dimensional superconformal field theories may be able to describe all ...


12

There is already a good answer by Solenodon Paradoxus. Here we provide a formal proof (via the stationary phase/WKB approximation). To fix notation, we define the effective/proper action $$ \Gamma[\phi_{\rm cl}]~=~W_c[J]-J_k \phi_{\rm cl}^k, \tag{1}$$ as the Legendre transformation of the generating functional $W_c[J]$ for connected diagrams. We assume ...


11

Although late in the party, I post an answer on an elemementary level. May be this proves the power of tensor calculus used in all previous nice answers. Abstract In this answer we'll try to derive Maxwell equations in empty space \begin{align} \boldsymbol{\nabla} \boldsymbol{\times} \mathbf{E} & = -\frac{\partial \mathbf{B}}{\partial t} \tag{...


11

Yes, there is a link, both are examples of extremum principles. And yes it is possible to derive the principle of least action from the law of maximum entropy. The derivation is lengthy and I will only sketch the main steps needed: We start from the law of maximum entropy $dS/dt \geq 0$. As we know this law is only valid for isolated systems [i]. For ...


11

The term vanished because we can translate this term to one making a statement about the fields at the boundary and assume that the fields themselves vanish in spatial and temporal infinity. By Stokes' Theorem, we can translate volume integrals into surface integrals. More specifically Gauss' Theorem states that the integral of a divergence of a field over ...


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